**Statistical Theory** * [Mathematics MSc Course -- Offered every Fall Semester]*

#### Course homepage | Short Outline:

- Stochastic convergence and its use in statistics: modes of convergence, basic convergence theorems.
- Formalization of a statistical problem : parameters, models, parametrizations, sufficiency, ancillarity, completeness.
- Families of models: the exponential family, maximum entropy principle, transformation families.
- Point estimation: the plug-in principle, Glivenko-Cantelli theorem, the moment problem
- Likelihood theory: the likelihood principle, asymptotic properties, misspecification of models
- Optimality: decision theory, minimum variance unbiased estimation, Cramér-Rao lower bound, efficiency.
- Testing and Confidence Regions: Neyman-Pearson setup, likelihood ratio tests, UMP tests, duality with confidence intervals, confidence regions, large sample theory

**Regression Models** * [Mathematics Senior BSc / MSc Course]*

#### Course homepage | Short Outline:

- Gaussian linear model: least squares, geometrical interpretation, distribution theory, Gauss-Markov theorem.
- Analysis of variance: subspace decomposition, Cochran's theorem, F-statistics, orthogonality.
- Model assessment: diagnostic plots, outliers, influence, leverage, collinearity, ridge regression.
- Variable selection: AIC, BIC, forward/backwards/stepwise procedures, the LASSO
- Generalised linear models: likelihood, iteratively weighted least squares, deviance, logistic and Poisson regression.
- Smoothing and generalised additive models.

**History of Mathematics** * [Mathematics First Year BSc Course]*

#### Course homepage | Short Outline:

- Order from chaos. Evidence on the origins of mathematical activity. The legacy of the Babylonians and Egyptians.
- The birth of the concept of proof. Pythagoras' theorem, before and after Pythagoras. The development of the axiomatic method. Euclid's elements and their ramifications. The birth of number theory. The legacy of the Greeks.
- The birth of algebra. Emergence of symbolic manipulation. Geometric approaches to cubic equations, and the legacy of the Arabs.
- The influence of astronomy. From static mathematics towards dynamic mathematics: the birth of the study of change.
- From certainty to uncertainty. The birth of mathematical probability.
- The inspiration of the infinitesimal and the birth of calculus. The fundamental theorem of calculus. Emergence of series approximations.
- Geometry revisited: the liberation of geometry and the non-Euclidean revolution.
- Algebra revisited: the liberation of algebra and the non-commutative revolution. The birth of group theory.
- Calculus revisited: the birth of analysis. The real number system's fundamental role. The emergence of set theory and abstract spaces. Grasping the infinite.
- The axiomatic method revisited. Can we prove or disprove everything? The incompleteness theorem.

**Introduction to Probability and Statistics** * [Environmental Engineering BSc Course]*

#### Course Homepage | Short Outline:

- Exploratory statistics (Data types; Graphical exploration of variables; numerical summaries of distributions; The boxplot; The normal distribution).
- Probability calculus (Probabilities of events; Random variables; Characteristic values; Fundamental theorems).
- Fundamental statistical concepts (Statistical models and parameter estimation; Confidence intervals; Statistical tests; Chi-square significance tests).
- Linear regression (Introduction; Least-squares principle; Simple linear regression; Multiple linear regression).

**Statistics for Data Science** * [Data Science MSc Core Course]*

#### Course Homepage | Short Outline:

- Probability Background (fundamentals, moment bounds, entropy, exponential families, random vectors).
- Sampling Theory (sufficiency, sampling distributions, stochastic convergence, Gaussian and exponential family sampling)
- Foundations of Statistical Inference (maximum likelihood, bias and variance, asymptotic efficiency, hypothesis tests and confidence intervals, nonparametric inference).
- Linear regression (Gauss-Markov theory, diagonstics, ANOVA, model selection, multicollinearity, L2 and L1 regularisation).
- Generalised Linear Models (Logistic and loglinear models, separation, asymptotic theory, contingency tables, Simpson's paradox).
- Generalised Additive Models (nonparametric regression, curse of dimensionality, projection pursuit, additive models, GAM, neural networks).